Gear



Feb. 4, 1941. E. wuJnl-lf-BER 2,230,418

GEAR 1 Filed Dec. 31. 1957 e sheets-sheet 1 Erge NiZof/Lb er ATTORNEY AFeb. 4, 1941, E. WILDHABER 2,230,418

` GEAR Filed Dec. s1, i937 e sheets-Sheet 2 Ermes? VViZchber INVENTOR BYe! ATTORNEYM/ Feb. 4, 1941. E. WILDHABER GEAR Filed Dec. 51, 19157@Sheets-Sheet 3 mlchcber |NVENTOR ATTORNEY Feb.4,1941. E;.W,LDHABER"2,230,418

GEAR

FiledrDeC. 51, 1937 6 Sheets-Sheet 4 INVENTOR ATTORNE Feb. 4,. 1941. E,WILDHABER 2,230,418

` GEAR Filed Dec. 31, 1937 6 Sheets-Sheet 6 INVENTOR ATTORNE toothedgears,

Patented Feb. 4, 1941 PATENT onen' Ernest Wildhaber,

Gleason Works, oi' New Yori:

nondequoit, N., y., assignmto Rochester, N. Y., a. corporationApplication December 3l, 1937, Serial No. 182,837

is claim'. (ci. 741-462) The present invention relates to gears and totheir manufacture, and particularly to straight e and to the manufactureof such gears.

selves to any variations in mountings and loads One purpose of lthepresent invention is to provide a tooth shape .for gearswhich will bevery easy to produce and which will permit of cutting' gears of the typedescribed on machines of extremely simple construction.

A kindred object of the invention is to pro-A straight toothed searswhose mating tooth surfaces will mesh with less than full prole contactand also with less than full length contact so that mate gears will runquietly when kin mesh and be capable', moreover, of adjustment relative;

to one another and readily accommodate themwhich may be met with in use.

Stili another object of the invention is to Drovide a tooth form forbevel gears which Vmay be milled or ground very easily and veryaccurately. A further object of the invention is to provide simple,accurate methods for producingl gears oi the character described eitherin'intermittent or continuous indexing processes.

Other objectsof Athe invention will be apparent hereinafterirom thespecification and from the recital o the appended claims.

According to the present invention, both gears of a pair of unequalgears are provided with tooth proles which are single, convex circulararcs or which closely approximate single, convex circular arcs.Moreover, the ratio of the radii of profile curvature of the teeth ofthe` smaller and larger Vgears isgreater than. the ratio oftherespective tooth numbers of the two gears, in the case of spur gears, oris greater than the'ratio of the tooth numbers of the respectivedeveloped complete back cones inthe case of bevel gears. The toothsurfaces of gears produced according to this invention may be eithercylindrical or conical or may closely approximate such surfaces.

Gears having the characteristics oi the present invention may beproduced in. various ways. Their tooth surfaces may be cut or groundwith a milling cutter or grinding wheel of ,circular or spur, helical,bevel, and hypoid,

position. There are of their pitch surfaces such Abevel gears, etc., aswell as to nearly circular prole by adjusting the cutter or wheel intofull depthengagement with the gear blank and moving the tool along thelength of a tooth. ,'Theirtooth surfaces may be cut with areciprocatingtool of circular or nearly circular c cutting profile by imparting acutting motion to the tool and producinga relative feed movement betweenthe tool and gear blank until the tool cuts to the full depth ot theteeth to be produced. Their tooth surfaces may be cut by l@ simplyimparting a cutting movement to a tool, While feeding it about thecenter of curvature oi the tooth profile to vbe produced upon the gearbeing out, until the tool has reached full-depth various other Ways,too, in"I 35 which they may be produced, as will appear hereinafter.

The invention may be applied equallyto gears whose teeth Vare inclinedto straight line elements as helicalgears, skew 2@ gears whose teethextend along the straight line elements of their pitch surfaces. such asspur gears, straight bevels, etc. In the case of gears with helical orskewv teeth, and broadlywhere the bottoms oi' the 3 toothspac of thegears are inclined to the straight line elements of the cylindrical orconical pitch surfaces of the gears, a motion about the axis ofthe toothsurface, which is to be produced,

is preferably provided simultaneously with the 80 motion of the toolalong the tooth being cut so that the tool may follow the desired toothbottom when it cuts the lowest part of the tooth dank. This additionalmotion does not alect the shape of the tooth surface proper but only theS5 A shape of the tooth bottom and does not therefore require the highaccuracy of the usual timed relationship. l vQrdinarily gears are cutaccording to the present invention in an intermittent indexing process.s@ The invention, however, is not 'limited to use in such a process. Ifdesired,vthe present invention may also be employed in the cutting ofgears according to a continuous indexing process where the tool isreciprocated along the length of the gear tooth yand the gear blank isrotated continuously in timed relation with the tool movement so thatthe tool cuts in a diierent tooth space of the blank on each cuttingstroke. Inl

such case, the desired tooth profile curvature may 50- suitablelocalized 'tooth bearing when they run A is employed on a machine of thecontinuous indexing type, may be retained, if desired, however. Thismotion permits of avoiding producing teeth whose lengthwise curvature isof 8 shape and at the same time enables the two members of a gear pairto be cut so that theywlll have a in mesh.

One advantage of gears produced according to the present invention overgears having unmodified or rigid involute tooth profiles is the presenceof a very desirable slight relief at the tops and bottoms of the toothprofiles. This minimizes the effect of manufacturing errors'and providest a gear which will run quietly with its mate.

' Another advantage is the reduction or elimination of undercut, whichoften exists especially on pinions of the involute system.

Still another asset is the better balance of the tooth profile curvaturebetween the pinion andv gear and the increased strength of the pinionteeth due to greater thickness at their bases.

,In the case of tapered gears and straight spur gears. the toothsurfaces are preferably portions of circular conical or circularcylindrical surfaces.

With such tooth surfaces, it is possible to make tapered gears, areportions of convex circular cylindrical surfaces represents a specialbut important embodiment of this invention. This modification of theinvention solves the heretofore .diillcult problem of devisingapractical tooth shape for bevel and hypoid. gears which will permitthese gears to be milled and accurately ground with very simple meansand without Vgenerating roll. In the case of gears made according tothis embodiment of the invention, all that is required is to provideamilling lcutter or a grinding wheel of circular arcuate profilecurvature and feed the cutter or grinding wheel along the tooth surface,that is, along the axis of the Ycylindrical surface, while moving thetool about said axis to cause the tip of the tool to follow the desiredtooth bottom. In many cases, moreover, the latter motion may be omitted.The present invention, then enables us to grind both 'spur and taperedgears very accurately with the most efficient grinding contact, namely,contact along the (circular) profile of the grinding wheel.

'I'his contact is analogous to the prole contact between grinding wheeland work obtained when grinding Formate spiral bevel gears according tothe known method, and which has proved extremely successful in practice.

In the drawings: .d

, Fig. 1 is Aan end elevational view of a pair of spur gears madeaccording to this invention, .or a prole Asectional view of a pair ofhelical gears made according to the invention or a view of the developedback cones of a pair of bevel gears made according to this invention;

`Fig. 2 is a corresponding view of a pair of gears of smaller pitch;

Fig. 3 isa diagrammatic view illustrating cer-l Y tain principles o!construction involved in a pair ,of gears made according to thisinvention;

Fig. 4 is a diagrammatic view illustrating one method of cutting gearsaccording to this invention and. illustrating the cutting of one side ofthe teeth and Fig. 5 isa corresponding view showing the cutting o`f theopposite side of the teeth of the .gear by the same method;

Fig. 6 is a diagrammatic view further illustrative of this method asapplied to the production of bevel gears; l

- Fig. 'l is a plan view illustrating the method of cutting a bevel gearaccording to a slightly different embodiment of the invention;

Figs. 8 and 9 are a fragmentary plan' view and side elevation,respectively, illustrating the application of the method shown in Fig. 7to the cutting of spur gears;

Figs. 10, 11 and l2 are diagrammatic fragmentary end elevational views,illustrating various ways of employing a pair of tools to. cutsimultaneously opposite side tooth surfaces of a gear by the process ofthe present invention;

Figs. 13 and 14 are fragmentary views taken at right angles-to oneanother and illustrating one way of grinding a ystraight bevelgeardaccording to the present invention;

Fig. 15is a plan view of a skew bevel gear made according to thisinvention and Fig. 16 is a view on an enlarged scale. of one of theteeth of this gear and illustrating diagrammatically certain of theprinciples of this invention;

Fig. 17 is a diagrammatic view showing a section taken midway the lengthof the tooth of Fig. 16-and in dotted lines anend view of the `tooth atthe large end thereof;

Fig. 18 is a diagrammatic view illustrating one method of cutting orgrinding the teeth of a skew bevel gear such as shown in Figs. 15 to 1'7inclusive; and

Figs. 19 to 2l inclusive are diagrammatic views illustrative of theprinciples of the invention, as applied to the embodiment illustrated inFigs. l5 to 17 inclusive.

Reference will now be had to the drawings for a more detaileddescription of the invention. In

Figs. 1 and 2, 25 and 26, respectively, denote the two members of a pairof gears made according to the present invention. As above stated, theseviews may be considered as the end views of a pair of spur gears madeaccording to the invention, in which case 21 and 28 denote,respectively, the axes of these gears, or these views may be consideredas the developed back cones of a pair of bevel gears made according tothe invention, in which case 21 and 28 denote, respectively, the apicesof the back cones. In the case `of bevel gears, then, the sphericalproblem is reduced to a plane problem with 21 and 28 as parallel axes ofrotation perpendicular to the plane of the drawings.

The gears are shown in a lposition where their mating tooth surfacescontact at a mean point P. The side surfaces 32 and 34 of the teeth 30of the pinion 2i are of convex circular arcuate prole and the sidesurfaces 33 and 35 of thel teeth 3i ,of the gear 25 are also of convexcircular arcuateproiile curvature. To transmit correct motion, thecommon proiile normal 36 at point P must pass through the pitch point Qwhich is the contact point of the pitch circles 31 and 38 of the twogears, and this common profile normal must contain the centers 39 and40, respectively, of the'mating proles 33 and 32, respectively.

a,2so,4is A YThe radius of profile curvature oi the tooth The tangent ofangle 40"-Q-'402 which angle surfaces oi the pinion Zlor smaller m berof' isdesignatedasis:

thepairatmeanpointPislai-gerthanth 'curvature radius ofthe prole of aninvolutetooth which has the same tooth normal at said mean peint. 50, asstated. is the center of proiile curvature oi tooth surface 32 oi apinion produced according to this invention whereas 42 would be thecurvature center ofthe proiile of an involute-tooth having the sametooth normal. The curvature center 42 Vof the involute prole is thepmiection oi the axis or center 28 tothe tooth normal 35. The differencebetween the radii oi curvature of aninvolute toothproiile and of a toothprole oi a pinion made according to the present invention increases withan increase in the ratio of tooth numbers of a pair of gears.-

On the gear, or larger member of the pair, 25.

the relationship between the radius of profile curvature and the radiusoi curvature of an involute prole is the reverse oi' that which occurson the pinion. The radius of tooth prole curvature on the gear issmaller thanrthe curvature radius of the proiile of a correspondinginvolute tooth. Thus, the center of curvature of the proiile 33 ina gearmade according to the present invention is at 3S whereas the curvaturecenter ci a corresponding involute proiile would be at d3, which is theprojection of the center or axis 2 of thegear 25 to the tooth normal 36.

The proiile radius P-ll of the tooth surface 32 of the pinion is made upof a distance Q-40 and of distance Q-P. The prole radius P-39 of thegear is made up of the diiierence betweenthe distance Q-SS and thedistance Q-Pf Let r and R denote, respectively, the pitch radii 28-Q and-Q of the two gears andV let denote the presure angle, that is, theinclination of the normal 3B to the tangent 45 of the pitch circles ofthe two gears at Q. Let r denote the distance Q-l and let R' Ydenote thedistance Q39. It is seen that:

also: distance (2S-l2) =r.cos qs.

'isY

When athe"; nion 2S turns on its center 28 through a small angle 0,profile center 48 moves on a circle E about the center 28 to new posiand the angle 6 should be equal distance (4W-40")I== distance (Q2-40')distance (4W-40") l g distance v((2-40)--distxstnce (4Q-40') tan gear 25on its center 25 through an angle 0' is:

distance (3c-39') =(R sin R {(R sin ad?) ots a'- distance (3W-39") Theangles 0 and 0.', respectively, through which the pinion and gear turnare-inversely' proportional to the pitch radii r and B, of the pai-r sotha' r i I Y The tooth proles of a pair of gears produced according tothis invention will fuliill the requirements of true uniform motionwhenthe tooth normal at the point 'of contact Iofthe mating toothsurfaces continues to pass through pitch point Q in any position ofrotation of the gears or also when the normal drawn to the pitch point Qperpendicular to the tdoth profile of the pinion coincides with thenormal drawn through Q perpendicular to the mating tooth profile of thegear. 'I'his'then means that for: angles of rota.-

tion 0 and 'I If 0.cienotes the arc of the angle, that irs-the.

\ angle in radians, we have Likewise for small angles,

f sui 0 -R 0 and r 2 (1 -cos 0) =5 0 Equations 1 and 2 then-become:

' tanA 15 The component 39-39' (Fig. 3) of the displacement of proiile'center 39 caused by turning the,

` tion then becomes:

Ii we now multiply both sides vof the equation by the two denominators,we obtain.

gem adds to the-cost or thema required to produce them.

.It lWe neglect terms higher than the second order in 0, we obtain: y

R 9S 41197] I l and Let us can' the mst term within the bracket C' andthe second term C". The above equaada-.geen

Both c' and c"' musi be zero to 11111111 An-approxiv equation for allsmall angles ofA 0. mation is obtained when only C' is zero and C" issmall or when C' is small and C" is zero, or

' when both c' and c" are man. vwhen amy c' 1s third order. thosefamiliar with infinitesimal quantities, the

departure of such proles from profiles giving a mathematically uniformmotion is in the same direction at both prolile ends. They have eithermorestock atthetopandbottom of the proiiles or they have less stock atthe top and bottom of the proles. I have found that it can readily bedemonstrated, that proilles of circular arcuate curvature, such as areproposed 'with the present invention, have slightly less stock at thetops and bottoms of the teeth, that is, they are slightly relieved atthe tops and bottoms of the teeth. This is just what is desired inpractice. In fact,` thismodiiication is so desirable that in practiceinvolute teeth and other teeth designed to trans-v mit uniform motionare commonly modied to have relief at the tops and bottoms of the teethalthough such modincation, in the case of such l and further,

. 1 If we let C'=zero, then the following equation is obtained whichresulta from dividing all the factors-by (1".R'):A

1` 1 1 1 .sm 4 [p+]=+ (3) Where C"=zero and the ifactors Iare divided by(r cos qb), then: I

. 2 r-f sin +r sin -R'Il} Rl+1 -rr=0 v If we multiply an the factors byn, we obtain;

is according to Equation 4v:-

R+2r=4f+2f g=g A 2R+r Sr-i-r-i) 3 whereas inthe involute system thecorresponding radii oi.' curvature ot the tooth surfaces of gear andpinion are in theproportion of V4;

` The quantities r' and R' are independent of the distance Q-R In otherwords, the locations of the prole centers 4l and 39. do not depend onthe position of the mean' point P along the normal 36. In fact, ifdesired, P may be so located conrmedthe results given above.

acercara along the normal 36 that equal profile radii Vmay be obtainedon the mating tooth surfaces of gear and pinion. The tooth thicknessesof gear and pinion maythen be balanced by providing a suitablylong'addendum on the pinion and'a'correspondingly short addendum on thegear.

Fig. 2 illustrates a pair of gears 25' and 26 which are of.finer pitchthan the gears shown in Fig. l, but which are of otherwise equaldimensions with the gears, shown in Fig. l. Equaladdenda are provided onboth gear and pinion in the pair shown in Fig. 2. The mean point ofcon--` tacts between the mating'tooth surfaces, which is at P in thepair shown in Fig. 1, is preferably assumed at pitch point Q in the pairshown in Fig. 2. It should be noted that the centers oi profilecurvature 49 and 39 have the same positions in both gures.

Bevel gears made according to this invention are treated according totheestablished practice in treating and analyzing lbevel gears, ,that is,like spur gears by using the back cone radii of the bevel gears as theradii r and R.. Accordingly, the back cone radii are first determined asusual and the locations of the centers of curvature of the toothproiiles of the bevel gears and the radii of curvature of these profilesare determined with Formulas 5.

The angles l and L included between the axis of the conical toothsurface of pinion or gear, respectively, and the straight line ofcontact between the conical pitch surfaces of the two gears when theyare in mesh is:

n T' tall tan L-A I where A denotes the cone distance, that isf, the

distance from the apex of Agear or pinion to the mean point of contact.

I have made a direct analysis of the spherical problem encounteredeinbevel gears and have The direct spherical analysis is more involved anda great deal longer than the plane one and is omitted here inasmuch asit merely conrms an expected result. I

Lengthwise mismatch or localization of tooth vbearing may be obtainedvin a pair of gears made according to this invention by the methoddescribed generally in myprior Patent No. 1,733,326 above mentioned. Thetooth surface produced on the two members of the pair have straight lineelements which do not coincide but which include a slight angle with oneanother. IheV gears so produced will have pressure angles which changefrom one end of the teeth to the other, but this is no drawback.

When the present invention is applied to the production of gears havinghelical teeth, the circular tooth profiles may be determined either inthe transverse (peripheral) section of the gears or in a section normalto the lengthwise direction of the teeth. The pressure angle 1: used inthe formulas already given Will,-then, be ther transverse or the normalpressure angle, respectively, in accordance with the piane in which itis desired to have the circular arcuate prole curvature. The radii r andR are the actual pitch radii of the cylindrical gears where thetransverse section is used andthe curvature radii of the normal sectionsthrough the pitch surfaces when the normal sections are used. v

All known methods Aof manufacturing gears may be used in `making gearsaccording to the present invention. Milling or grinding with cutters orgrinding wheels of circular profile is particularly simple when theprofile of the tooth surface to be produced is constant along the lengthof the tooth surface and need not be ex plained further here. It will beobvious that when a cutter or grinding wheel of this type is employed,the desired constant tooth profile curvature can be produced on the gearbeing Vcut or ground by simplel movement oi the cutter along the geartooth.

Figs. 4 and 5 illustratethe application of the planing method ofncuttingto the production o gears according to the present invention. Eitherspur or bevel gears may be cut in this way. The two sides of the teeth,of the gear are cut separately. Y

t@ designates the gear to be cut and 5l is its axis. 52 is a tooth ofthe gear and 53 and 5d are the opposite sides of the tooth. 55 is theplaning tool for cutting one side 53 of the teeth and 55 is the planingtool for cutting the oppo- 'I site side 5d of lthe teeth. These toolshave straight-sided cutting edges, `but may be rounded,

as shown, at their tips according to conventional construction.

The tooth surface 53 to be cut is of circular arcuate prole shape andhas its center at 5i and a radius of curvature 59. The opposite sidetooth surface 5d has a center of curvature at tt and a radius ofcurvature t2.

In cutting the tooth side 53. the gear is held stationary on its axis'5i and the planing tool is reciprocated back and forth along the lengthof the gear tooth while a slow relative feed movement is producedbetween the. tool and the gear blank about the axis 5l of profilecurvature-oi' the tooth surface to be cut. In Fig. 4. this feed movementis illustrated as applied to the tool so` that during the cuttingoperation, the`tool moves from the full line position indicated at 55 tothe dotted l-ine position indicated at 55'. When the tool has reachedfull depth position, the tooth side 53 is completed. The tool thenisswung back about the axis 51 until it clears thel blank and then theblank is indexed.

Afterall of the sides 53 of the teeth of the blank' tooth sides are thencut with the tool 56 in the 'same Way as the sides 5i! were cut, namelyby reciprocation of the'tool along the length of the gear tooth andrelative feed of the tool about the profile center or axis tt. n

In planing the teeth oi' either a spur or a bevel gear, the planing toolis, of course, recipro-I cated inthe direction of a straight lineelement of the tooth surface being produced. For bevel gears, thestraight line elements of the tooth surfaces ordinarily intersect in thecone apex of the gear. Thus, as shown in Fig. 6, in planing the toothsurface of the bevel gear 66, the planing tool 61 will'ordinarily bereciprocated along lines 68 which intersect in the gear apex 69. 10 isan element of thetooth surface at the top of the tooth profile and 'H isan element of the tooth surface at the root of the tooth prole. yThetool is shown in full lines at 61 at one end of its stroke and in dottedlines at 61' at the opposite end of the cutting stroke. By reciprocatingthe planing tools along lines differently inclined tofthe pitch surfacesofthe mating gears, however., in a manner similar to that described inmy patent above mentioned,

mating tooth surfaces can be produced on a pair of gears which willmismatch one another and so a localization of tooth bearing may beobtained. 'The planing process of cutting gears has the advantage ofextreme simplicity and a very sim- `ple form of gear cutting machine maybe emhave been the requirement for a templet to control the tooth shapeand the fact that the cutting has had to be done with a pointed. toolora tool with a small round at its tip. Templets are, of course,difllcult to produce with accuracy 'l5 the toothvproiiletobeproduced.Hence, with a consists of a motion such thatthe two tooth l and thecutting action itself with the pointed tool gives a poor tooth surfacefinsh. Moreover, it is not very accurate and the cutting process withsuch a tool is relatively slow.

With the present invention, however, the drawbacks of the planingprocess are all overcome. The machine, in fact, can be made even simplerthan prior types of gear planing machines, for to out the gears of thepresentinvention, the reciprocating tool needs only to be fed about afixed axis, the axis of profile curvature of the tooth surface, whereasheretofore it had to be guided by a templet. In addition, the toothprofiles of the gears of this invention can be cut with tools havingstraight cutting edges and in this way the speed and accuracy of agenerating process can be obtained. It is possible to do .even more thanthis. By using a planing tool that has a-concave cutting edge, a bettertooth surface can be obtained than in a generating process or the samefinish may be secured but in a shorter time. The-use of tools havingconcave cutting edges is illustrated in Figs. '1 to 9 inclusive.

Fig. 'I shows the cutting of one side surface of a tooth of a bevel gearwith such a tool. The tool is denoted at 1l and it has a concave cuttingedge 10. 11 denotes the gear and :18 isits apex.

1l is the axis of the conical tooth surface le' which is to be produced.In cutting the gear, the tool is reciprocated in direction 82 so thatits point-of contact Il moves along a line 8l radial of the gearapex'il, and simultaneously with the reciprocating movement, -the toolis fed gradually about the axis 19 into full depth position. The toolpath 8l includes an angle c with the axis 1l ofethe surface beingf-cut.This angle varies, of course, for different gears, but .a machine may bemade to cut different gears simply by providing an adjustmentofthe toolabout gear apex 1l.

The use of a planing tool having a concave cutting edge is furtherillustrated in Pigs. 8 and 9 in connection with the cutting of a spurgear. Here 8l denotes the gear to be cut and II is its axis. I1 is thecutting tool and 88 denotes the concave cutting edge of-'this tool; l!is the tooth surface which is to be cut with the tool 81. The tool isshown in contact with this tooth surface at the point il. The axis ofprofile curvature of the tooth surface 8l is denoted at $2. The radiusof curvature II-ll of the cutting edge Il of the tool I1 is larger thanthe radius "-01 of pronlecurvature af the tooth surface l! to be cut.It-

will be evident, however, that the concave cutting edge of this toolwill follow the convex tooth profile 8l better than a straight sidedcutting tool because the concave cutting edge differs less fromteoinsving s eeneave cutting edge. a better tooth surface nnish may beobtained.

In cutting the gear Il, the tool is reciprocated.

'as before, in the direction or tooth length, neredenotedatllandisfedaboutthe axis!! of 5- proille curvature of the teeth.Here the direction Si is parallel to the axis l2 and, for spur gears, isalsoparallel to the gear axis I8. The

tooth surface cut onpthe gear. then, is a cy1in drical surface whoseaxis is at 92. After one tooth surface of the gear has been cut, thetool is withdrawn and the blank indexed. After all the-tooth Y surfacesat one -side of the teeth have been cut,

the opposite sides of the teeth may be cut 'with a tool having a cuttingedge suitable for cutting such sides.

In a helical gear, the teeth are, of course, inclined to the gear axis.In Fig. 8, we have shown diagrammatically how a helical gear may beproduced. The teeth of this gear are assumed to extend in the direction95 and the'axis of the gear Ais assumed to have the position indicatedby the dotted line Si. 'Ihe cutting motion employed in cutting a helicalgear differs from that usedfor cutting a spur gear because of theinclination 25 of the teeth of thehelical gear to its axis. The cuttingmotion, for helical gears, should be a helicoidal motion, -as denoted bythe arrow", about the axis $0 of the gear. This means that the tool mustbe reciprocated in the direction of 30 the axis 96 while the gear isbeing turned about that axis or that the tool may be held stationarywhile the work is being reciprocated in the direction of and rotatedabout the axis Il. In addition to this helicoidal motion about axis arelative feed movement is produced between the 'tool and gear about axis02 to produce the desired profile curvature of the gear tooth, asbelhelical and tapered gears according tothis invention. In Fig. 10, IIIdenotes the gear to be cut and Ill and il! are the vtools which areelnployed to cut the tooth surfaces of this gear. The tools here shownhave concave cutting edges III and I, respectively,whose centers ofcurvature are-at lll` and I", respectively, and whose radiiof curvatureIl1-Ill and ile-ill, respectively, ,Bre greater than the radii ofcurvature lll-lil and lll-III, respectively,of the tooth surfaces' IIIand H2, respectively, which the tools are to cut. In Fig. 10, the'twotools are shown adjusted to operate on opposite sides III and Il! of thesame tooth space. Here the-two tools are ar- 55`rangedsoastocutinoppodtedirectionaonetool being slightly withdrawn forclearance while the other one is cutting. vOne tool cuts on the strokein'one direction and the other on the return lstroke and the toothpronles are produced, as 00 before, by reedinguie reeiproeeung rooie um:the axes-lll and III, respectively,l of profilev curvature of the toothsurfaces to be produced.u

ment with the blank andthe blank indexed.

Fig.11showsasetupinwhichthetwotools operate in different tooth spaces ofthe gear blank being cut. The blank is here denoted at 7 illand thetools at Ill and H1, respectively.

Here the tools have straight cutting edges' ill and Ill, respectively.With such wols, the feed motion is preferably imparted to the gear.' It

' andere sides |20 and |21, which are to be cut, will fremain in contactwith the stationary plane surfaces ile and HS represented `by the toolters coincide with theoaxes |22 and 23, respectively,lof prole curvatureof the tooth surfaces |20 and |2l to be produced. The abutments willhave plane surfaces representing the planes H3 and H3 or planes parallelthereto. The cams will be securedto the gearxin such way as to bearagainst the abutments. By rocking the gar about its axis, then, as thetools are reciprocated back and forth in the planes litl and H3,respectively, tooth surfaces |20 and |2| of the desired prole curvaturecan be producedpn/ the gear blank. When a pair of tooth surfaces hasbeen completed, the tools are Withdrawn from engagement with the blankand the blank indexed. Instead of reciprocating tools, plane-sidedgrinding wheels may be used, as will readily be understood.

'I'he motion obtained in cutting a gear by the method illustrated inFig. 11 lis actually a rolling `motion and it can readily bedemonstrated that on a spur gear, the motion is as if a circle of thegear would roll on a stationary circle twice its diameter. The rollingV,circle of the gear passes through the centers |22 and |23 of the prolecurvature and the point |25 which is the point of intersection of thetooth normals.

In Fig. 12, I have illustrated particularly a` cate back and forth inthe positions shown, the

blank is rst turned slowly about its axis |35 toward' the tool |3| untilit reaches the position shown in full lines. In this position, the point|36 at the root of the tooth prole |31 is formed. The blank is now heldstationary on its axis and the tool feed movement starts. In theinstances shown, the tooth profiles |31 and |38 are both curved about acommon axis |39. The feed movement of the tools is, then, of course,about this axis. This feed movement is such that as the tool 3| movesoutwardly to finish. cut the profile |31 from the root to the tipthereof, the tool |32 will move inwardly to finish-cut the profile |38from the tip tothe root thereof. At the end of the feed movement, thetool |32 will have reached full-depth position and the tool |3| willhave cleared the work. .The reciprocating movements of the tools arethen stopped, theitools are returned to starting position and the blankis indexed. The indexing motion is so retarded that the teeth of theblank, after indexing, will be in a position corresponding to that shownin dotted lines so that afterthe tools start to cut on new The toolshaveconcave cutting on. the center line of a tooth or of a tooth spaceof the gear. This requirement is ey et, however, by standardizing ongears of suitable Vtooth numbers. f

It will be obvious that the various modifications of the invention'above described may be practiced by using milling cutters or grindingwheels in place .of the planing tools. tools having cancave cuttingedges, then, milling cutters or grinding wheels having concave cut- Ating profiles may be employed and for the plan- For the planing ingtools having straight cutting edges, milling t along the length of thegear tooth is required.`

The feed movement about the axis M2 ofthe conical or cylindrical toothsurface d3 which is to be ground is in this instance preferably impartedto the gear blank |35. `Twol positions of the gear tooth are shown infull lines and in dotted lines, respectively, at the beginning and nearthe end of the feed movement about axis ldZ. In the position shown infull lines, the axis of the gear is at MG During grinding, the gear isswung about the axis H12, which is perpendicular to the draw-` ingplane, to and beyond the position'shown in dotted lines. The gear axismoves, in this swinging motion' on circle |31 to and beyond position966|. Y

In order to distribute the wear on the grinding wheel more evenly and tospread it over a larger area, the wheel may be moved in the plane of itsactive surface during the feed movement. Thus,`

it may be moved from the full line position shown to the dotted lineposition denoted at W' in Fig.

13. This movement may be timed to movement of the gear about axis m2.Where there is considerable stock to be ground dil off a tooth surface,a. tooth may be fed back and v forth over the grinding wheelA several'times before it is indexed and, if this is done, the gear may be Yadvanced step by step into the grinding wheel after each feed movementaboutthe axis id2, by slight rotation of the gear. on its own axis Mit.This advances the tooth surface to be ground step .by step into thewheel to permit step by step grinding oii' of the desired amount ofstock. After a tooth surface has been ground, the gear may be swungclear of the wheel and indexed.l

' Figs. 15 to 17 inclusive illustrate an application of the invention toskew bevel gears. In Fig. l5, I, have shown a skew bevel gear |50. Theaxis of this gear is denoted at |5| and its teeth are designat'ed at|52. 'I'he opposite sides of the teeth of this gear are of circulararcuate curvature and are preferably cylindrical surfaces so that theymay readily be milled or ground.

In Fig. 16, a tooth |52 of the gear is shown on an enlarged scale. |54denotes the pitch line on side surface |53 of the tooth and |55 is amean point in the tooth surface. |56 denotes a tangent tothe pitch line|54 at mean point |55. |51 denotes the normal projection to the toothsurface of the instantaneous axis of rotation of the' |51 is also thegear when in mesh with its mate. line of instantaneous contact betweenthe lmating tooth surfaces of a pair 'of fully matched skew straightline element ot the mating surfaces. Preferably, however, a pair of gars made according to this invention are provided with a slightlengthwise mismatch or localization of tooth bearing in accordance withthe principles of my prior patent already mentioned. When the gears aremade with this mismatch, the tooth surfaces lwill havepressure angleschanging along pitch line |54 from the large to the small end of theteeth.

This change in pressure angle is illustrated diagrammatically in Figs.19 and 20, which are views of the tangential planes to the surfaces ofthe mating gears at mean point of contact |55. Fig. 19 shows thetangential plane for the tooth surface of the gear shown in Fig. 16 andFig. 20 is the tangential plane of the mating tooth surface of the gearwhich meshes with gear |50.

' Io obtain the desired localization of length- 'wise tooth bearing, thetooth surfaces of the gear lil are so cut by lengthwise movement of thecutting tool that a straight line element |59 ,of a tooth surface isinclined to the pitch line tangent |56 at an angle which is greater bythe angle w `than the angle W of inclination of the instantaneous axis|51.to the tangent |56. 'I'he mating gear is then cut so that a straightline element |80 of one of its tooth surfaces is inclined to the tangentIIB at an angle which is less than the angle W by an angle W which may or may n'ot, as desired, be equall to the angle 1d'.

The rate of increase in pressure angle from the large to the small endof the gear teeth may read'- ily be computed by known means. I et` ,idenote the pressure angle at the mean cone distance A (distance isi-lil,t denote the spiral angle or lengthwise inclination of the teethof thegear at mean cone distance A and let o and w', respectively. denote thepressure angle and spiral angle at'any other cone distance |5||| (Fig.21). Letfybethe pitch angle ofthegear'Illandlbetliepitchangleofthematinggear.

The curvatureradii r1 and rx ofthe normal sections of involute teeth areknown to be at the pitch lines: v

l 1:22'. ,.1 ,I A, sin *,(cotann-i-cotanl) tanl".

This sum ofthe ofthe curvature radii of a normal section is independentof Athe .natureofthetootmasiswellknowninthearh and applies also tonon-involute teeth. A constant' sum and constant individual curvatureradii result when the factor.

is constant al1-along the length of the gear,

The tooth surfaces may then be made'cylindr'lcal` surfaces.

The rate of change o f oosfilf A.7 sin 7 and aise ef Alsinl should beaero for an infinitesimal distance s, such diens and there c is aconstant which may be determined in known manner from the showing (A-i-s'cos up) (sin C.s cos A sin s 2 sin .c.cos cody eos costa cosEvidently the term within tneperen'theeis must beaerosince Atina 006W'isaconstant. Hence s sie s) cosvlf o When'the above relationshipdsfulfilled,'we;

may use cylindrical tooth surfaces on the mating skew bevelgears. Theabove formula gives the constant C in terms of the spiral angle or toothinclinationi/andtl/iepressureangleaoritpermits of computing fthe spiralangle or tooth inclinationrpwhenaandchavebeenassumed.'

In Fig. 17. I have shown av section taken midway of the skew gear toothlli and superimposed upon a dotted line view of the'large end oi' theVtooth. The axis-of the cylindrical side surface Since the`straightlineelement |81 of the tooth surfa'is'inclinedtothepitehsurfaceofthegeantheaxisflllwill'alsobeinelinedtothepitchlineandat' l the toOth is indicated atIll.

the agigie-r'eqmredtoproduce atoothsurface f imvimr' the desiresleeaimuen er teeth bearing.

This construction is illustrated in Pig. 17, for itwillbeseenthatthetopofthetoothatthe middle of the tooth extends-furtherahovethe straight line element II'I of the tooth surface than does thetop of `the tooth at the large ena of the tooth.

Skew bevel gears made according tc [this inven-r is provided with aconcave operating surface |68 Y whose radius I 59|65 corresponds to theradius of the cylindrical tooth surface to be ground. In cutting orgrinding the gear, the cutter or grinding wheel is rotated on its axis|61 and moved longitudinallyofthe tooth surface and simultaneouslytherewith is rocked about the axis |65 so that, as it moves from thesmall to the large end of the tooth, it will cut or grind the tooth tothe desired depth. In Fig. 18, two positions of the cutter or grindingwheel are shown. The full line position is the position which the cutteror wheel occupies when it is grinding midway the length of the geartooth. The dotted ,line position, denoted at |66', is the position whichthe cutter or grinding wheel occupies when grinding at the largey endoi' the tooth, that -is, the position cor--A responding to the dottedline position of the tooth shown in Fig. 17. As before, when a toothsurface has-been ground, the cutter or wheel is withdrawn fromengagement with the blank and the gear is indexed and when one side ofall the teeth of the cutter have been ground, the wheel or cutter andthe blank can be readju'sted to permit grinding of the opposite sides ofthe teeth'.

In another important modification ofthe invention, tapered gears areformed with skew teeth which have lengthwise mismatch or localization oftooth bearing and which have conical side tooth surfaces whose axes areso located that the cone element, which is at the lowest point on theside of. the tooth surface,I follows the direction of the tooth bottom.The tooth surfaces of such a gear can be produced without rocking thecutter or grinding'wheel about the tooth surface axis as it moves fromone end of the tooth to the other and as is required in the processillustrated in Fig. 18. In the modication of the invention here referredto, the mean element o f the conical tooth surface is disposed to followthe pitch line tangent |56 (Figs. 16 and 19) more closely than the line|51, Angles w' and W' (Figs. 19 and 20) are then assumed to be .negativeandof the same order as W. The tooth surface produced on a gear made inthis way. has pressure angles decreasing from the-large end to the smallend of the tooth and the spiral angle or inclination ofthe teeth ismoderate, being 10 or 12, at most. This design of teeth for bevel gearslendsitself especially to the production of the gears in a planingprocess. The tool, whose cutting edge may be either straight or `concave'is reciprocated along the length of thesgtooth and is i'ed relative tothe gear blankahut theaxis of thel conical surface whichit is desired toproduce on the tooth. This axis is, of course, inclined to the directionof longitudinal movement of the tool at the cone angle of the toothsurface.

While the'invention has been described specically with reference togears whose tooth proles are of circular arcuate curvature, -it-isapplicable also to gears having tooth profiles which are notexactlycircular, particularly when such tooth profiles are nearer to a circulararc in curv ?5 vature than to an involute. In this case, the

ofthe gear blank is cut on each revolution of the mean radii ofcurvature r and R' are between the values given by Equation 5 for purelycircular profiles and the values for involutes` which are: r'=r sin andR=R sin c. They are the nearer tothe values given by Formula 5, the 5closer the tooth profiles are to circular arcs. The curvature radii,that is the radii from the pitch point to the curvature center,moreover, should fulfill Equation l3. f

The 'invention is applicable also to gears pro- 1o duced by theRevacycle process, as disclosed in my pending applications, SerialNumber 181,177, filed December 22, 1937, and Serial Number 182,838,filed December 31, 1937. In this process, a rotary vdisc cutter isemployed that has a plu- 15 rality of cutting blades arranged part-wayaround its periphery with 'a gap between the last and iirst blades and atooth side of a tooth space cutter.

Tapered gears can, moreover, b e produced with exact cylindricalsidetooth surfaces, using the Revacycle" process. In this case,thecorresponding side-cutting edges of all of the finishing blades ofthe cutter have the same radii of curvature 25 and the centers ofcurvature of the successive corresponding side-cutting edges are atprogressively varying distances from the axis of the cutter. -Successiveblades are simply made of progressively-diierent heights, 'the :finishcutting 30 tion following, in general, the prlnciplesof the 4b inventionand including such departures from the present disclosureias come withinknownor customary practice in the art to which the invention pertainsand as may be applied to the essential features hereinbefore set forthand as fall 45- within the scope of the invention or the limits of theappended claims.

" Having thus describedmy invention, `what I claim is:y l

1. A pair of unequal gears for tranmitting 50 uniform motion in which"the ratio of the effective curvature of the mating tooth profiles ofthe smaller and larger gears is greater than the ratio of the toothnumber of the gears.

2. A -pair of unequal gears `for transmitting 55 uniform motion havingteeth which are longitudinally\ straight4 in development and toothsurfaces which change in pressure angle from one end of e teeth to theother, the profiles of said teeth being, for the full operative heightof the 60 teeth/single convex circular arcs.

3. A pair of tapered gears for transmitting uniform motion, each ofwhich has ltooth surfaces-which are portions of single convexcylindrical surfaces. whose axes extend longitudinally 65 of the teethand whose straight line elements are inclined to the pitch surfaces ofthe gear.

4.1A pair of tapered gears for transmitting uniform motion, each ofwhich has tooth surfacesfwhic'h are portions of single convex coni- 70cal surfaces whoseaxes extend longitudinally oi' the teeth.

5. A pair oi' straig t toothed gears for transmitting uniform mc ionwhose side' tooth surfaces are crowned longitudinally and, for the fullI operative height of the teeth. are of single circular arcuate .proleshape.

6. A pair of straight tooth gears each of which has side tooth surfaceswhich, for the full operative height of the teeth,-are single convexcircular arcs, the opposite sides of spaced teeth of each gear havingthe same center of curvature.

7. A pair of straight toothed gears for transmitting uniform motion,each of which has teeth whose profiles-are single convex circular arcs,mating tooth surfaces of the lgears 'having straight line elements whichare inclined.l to one another.

8. A pair of imequal gears for transmitting motion, each of which hascylindrical side tooth surfaces whose profiles, for the full operativeheight ofy the teeth, are single convex circular arcs, the pressureangles of the tooth surfaces of both members of the pair changing fromone end of the teeth to the other.-

9. A gear having longitudinally inclined teeth whose side tooth surfacesare of circular arcuate profile and conical.

10. A pair of mating gears for transmitting uniform motion, each ofwhich has side tooth surfaces which, for the full operative height ofthe teeth, are single convex circular arcs, the

ratio of profile curvature of mating tooth surfaces of the two gearsbeing greater than the ratio of the tooth numbers of the gears.

11. A gear having teeth which are longitudinally straight in developmentwhose side surfaces are of constant single circular arcuate proille`curvature along their lengths, the axis of curvature of a tooth surfacebeing inclined to the pitch line of the surface.

12. -A straight toothed gear having teeth whose active tooth surfacesare portions of single cylindrical surfaces, the axis of curvature of atooth surface being inclined to the pitch surface of the gear.

13. A gear having longitudinally straight teeth whose active toothsurfaces .are portions of single conical surfaces. the axis of curvatureof a tooth surface being inclined to the pitch surface of the gear.

14. A pair of tapered gears having longitudinally inclined teeth whoseside tooth surfaces are conical and of single convex circular arcuateprofile shape, the pressure angle of said tooth surfaces decreasing fromthe large to the small ends of the teeth. r

15. A tapered gear having teeth which are longitudinally straight in'development4 and which have side tooth surfaces that are portions ofconvex cylindrical surfaces whose straight line elements extend in thedirection of the length of the teeth.

16. A tapered gear having teeth which are longitudinally straight indevelopment and which have side tooth surfaces that are portions ofcongical surfaces', the axis of each conical tooth sur- .face beingapproximately parallel to the bottom

